A TOPOLOGICAL EXISTENCE PROOF FOR THE SCHUBART ORBITS IN THE

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A TOPOLOGICAL EXISTENCE PROOF FOR THE SCHUBART ORBITS IN THE

A TOPOLOGICAL EXISTENCE PROOF FOR THE
SCHUBART ORBITS IN THE
COLLINEAR THREE-BODY PROBLEM
RICHARD MOECKEL
To Carles Simó on his 60th birthday – molts anys
Abstract. A topological existence proof is presented for certain symmetrical
periodic orbits of the collinear three-body problem with two equal masses,
called Schubart orbits. The proof is based on the construction of a Wazewski
set W in the phase space. The periodic orbits are found by a shooting argument
in which symmetrical initial conditions entering W are followed under the flow
until they exit W. Topological considerations show that the image of the
symmetrical entrance states under this flow map must intersect an appropriate
set of symmetrical exit states.
1. Introduction
The collinear three-body problem describes the motion of three point masses
moving on a line under the influence of their mutual gravitational attraction. It
was first studied by Euler [6] who was able to find simple explicit solutions – the first
solutions of the three-body problem ever found. Euler’s solutions are homothetic;
the configuration formed by the three masses shrinks while maintaining the same
shape. The solutions end in triple collision.
It is intuitively clear that double and triple collisions will play an important
role in the problem. It is possible to regularize the double collisions so that the
colliding particles bounce and then move apart. However, McGehee [8] showed
that there is no such regularization for the triple collision. Instead, he devised
a system of coordinates and a change of time scale under which motions which
previously ended in triple collision at some finite time, now approach an equilibrium
point as the rescaled time tends to infinity. By combining the regularization of
double collisions with the blow-up of triple collisions one obtains a dynamical system
without singularities. It is this modified collinear three-body problem which will
be studied here.
The goal of this paper is to give a topological proof of existence of certain simple,
symmetric periodic motions in the case where two of the three masses are equal.
The behavior of these orbits is shown in figure 1. Initially, the unequal mass, m3 ,
is at the midpoint of the equal ones. This is Euler’s central configuration. If the
particles were released with zero velocity, a homothetic collapse to triple collision
Date: June 6, 2006 (Preliminary Version).
2000 Mathematics Subject Classification. 70F10, 70F15, 37N05, 70G40, 70G60, 70H12.
Key words and phrases. Celestial mechanics, three-body problem, symmetrical periodic solutions, topological methods.
Research supported by NSF grant DMS 0500443.
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RICHARD MOECKEL
1
Figure 1. Symmetrical periodic solution for m3 = 10
, 1, 10. The
orbit are collinear but center of mass has been given a small downward drift.
would ensue. However, the initial velocities are chosen so that m2 and m3 approach
one another while m1 moves in the opposite direction. The main point of the paper
is to show that it is possible to choose the velocities so that m1 slows to a stop
at exactly the same moment when m2 and m3 collide at u = π2 . This represents
the first quarter period of the periodic orbit. During the next quarter period, the
particles return to the initial configuration but with the velocities reversed. Finally,
this sequence is repeated on the other side with m1 and m3 colliding. Such orbits
were found numerically by Schubart [10] (see also [7]). The author is not aware any
previously published existence proof, however.
Perhaps of greater interest than the orbits themselves is the method of proof,
which is a variation of an idea used by Conley in the restricted three-body problem
[1]. In Conley’s paper, the retrograde lunar orbit of Hill is shown to exist for a wide
range of values of the Jacobi constant. After regularizing double collisions, the
problem becomes one of finding a solution of a system of second-order differential
equations in the plane which moves from the positive x-axis to the positive y-axis
across the first quadrant and meets both axes orthogonally.
Such a solution is found by a kind of shooting argument. Points starting orthogonal to the x-axis are followed until one of two exit conditions holds – they either
hit the positive y-axis or their velocity vectors become horizontal. As the initial
point along the x-axis varies, the final behavior changes from hitting the y-axis
with nonzero slope to having a horizontal velocity vector before hitting the positive y-axis. Somewhere in between, there must be a point whose velocity becomes
horizontal exactly when it reaches the positive y-axis and this gives the desired
periodic solution. The main difficulty in justifying such an argument is to show
that the initial solutions really arrive at one of the two kinds of final states, and
that the final state depends continously on the initial condition. For this, Conley
constructed an isolating block.
COLLINEAR PERIODIC ORBITS
3
Isolating blocks were developed by Conley and Easton as a way to defining a
topological index for invariant sets [2, 3, 5]. Among their useful properties is the fact
for initial conditions which leave the isolating block, the amount of time required
to leave depends continuously on initial conditions. It follows that the location of
the exit point also varies continuously. In [1] Conley constructs an isolating block
in a manifold of fixed Jacobi constant and uses it to justify the shooting argument
outlined above.
The concept of isolating block is related to earlier ideas of Wazewski [11]. It
is possible to get the crucial property of continous exit times under weaker assumptions than are needed for the topological index theory. For example, whereas
isolating blocks are always compact, Wazewski sets need not be. In this paper the
proof will be based on the construction of a Wazewski set, rather than an isolating
block. It is hoped that Wazewski sets and higher-dimensional shooting arguments
will provide existence proofs for symmetrical periodic solutions in more complicated
systems, such as the planar three-body problem or symmetrical subsystems of the
n-body problem.
2. Equations of Motion and Regularization
Consider the collinear three-body problem with two equal masses m1 = m2 = 1
and an arbitrary mass m3 > 0. Let the positions be qi ∈ R and the velocities be
vi = q̇i ∈ R. The Newton’s laws of motion are the Euler-Lagrange equation of the
Lagrangian
L = 12 K + U
(1)
where
K = v12 + v22 + m3 v32
m3
m3
1
+
+
.
U=
r12
r13
r23
(2)
Here rij = |qi − qj | denotes the distance between the i-th and j-th masses. The
total energy of the system is constant:
1
2K
− U = h.
Assume without loss of generality that total momentum is zero and that the
center of mass is at the origin, i.e.,
v1 + v2 + m3 v3 = q1 + q2 + m3 q3 = 0.
Introduce Jacobi variables
ξ1 = q2 − q1
ξ2 = q3 − 21 (q1 + q2 )
and their velocities ηi = ξ˙i . Then the equations of motion are given by a Lagrangian
of the same form (1) where now
(3)
K = 12 |η1 |2 + µ|η2 |2
1
m3
m3
U=
+
+
r12
r13
r23
4
and µ =
RICHARD MOECKEL
2m3
2+m3 .
The mutual distances are given by
r12 = |ξ1 |
(4)
r13 = |ξ2 + 12 ξ1 |
r23 = |ξ2 − 12 ξ1 |.
The use of Jacobi coordinates eliminates the translational symmetry of the problem
and reduces the number of degrees of freedom from 3 to 2.
The next step is to replace (ξ1 , ξ2 ) by variables representing the size and shape of
the configuration. The approach used here is the collinear version of the Hopf-map
reduction of the planar three-body problem [9]. Define r, w1 , w2 via:
r2 = 21 ξ12 + µξ22
(5)
w1 = 14 ξ12 − 12 µξ22
q
w2 = µ2 ξ1 ξ2
It is easy to check that the new variables satisfy the relation:
w12 + w22 = 41 I 2 = 14 r4 .
The quantity I = r2 is the moment of inertia, a convenient measure of the
overall size of the configuration formed by the three bodies. Its shape can be
represented by the normalized vector s = 2w/r2 which lies in the unit circle S1 . It
will be convenient to replace s by an angle θ such that s = (cos θ, sin θ). The angle
describes the shape of the collinear configuration. The shapes which are relevant
here are those for which the mass m3 lies between the equal masses m1 , m2 . This
corresponds to the interval −θ∗ ≤ θ ≤ θ∗ where
p
θ∗ = tan−1 m3 (2 + m3 ).
It is easy to see that θ∗ increases from 0 to π2 as m3 increases from 0 to ∞.
The variables r, θ satisfy the Euler-Lagrange equations of the Lagrangian
(6)
L = 12 K + 1r W (θ)
where:
(7)
K = ṙ2 + 14 r2 θ̇2
m3
m3
1
+
+
.
W =
ρ12
ρ13
ρ23
Here ρij = rij /r is the normalized interparticle distance, i.e., the interparticle
distance after scaling the configuration to have moment of inertia 1. After some
computation using (4) and (5) and some trigonometric identities, one finds
ρ212 = 1 + cos θ
(8)
ρ213 = A sin2 ( 21 (θ + θ∗ ))
ρ223 = A sin2 ( 21 (θ − θ∗ ))
where
A=
1 + m3
.
m3
COLLINEAR PERIODIC ORBITS
5
The corresponding second-order Euler-Lagrange equations are:
r̈ = − r12 W + 14 rθ̇2
(9)
˙
r2 θ̇ = 4r Wθ .
Next, one can blow-up the triple collision singularity at r = 0 by introducing
3
the time rescaling 0 = r 2 ˙ and the variable v = r0 /r [8]. Setting τ = θ0 gives the
following first-order system of differential equations:
r0 = vr
v 0 = W (θ) − 21 v 2 + 2rh = 14 τ 2 + 12 v 2 − W (θ)
(10)
θ0 = τ
τ 0 = 4 Wθ − 12 vτ
with energy equation:
1 2
2v
(11)
+ 81 τ 2 − W (θ) = rh
Note that {r = 0} is now an invariant set for the flow, called the triple collision
manifold. The differential equations are still singular due to the double collisions at
θ = π, ±θ∗ . Since m3 is between m1 and m2 , only the double collisions at θ = ±θ∗
will be relevant. The final coordinate change will eliminate these singularities.
Define new variables u, γ such that
θ = θ∗ sin u
γ = τ cos u.
Note that −θ∗ ≤ θ ≤ θ∗ corresponds to − π2 ≤ u ≤ π2 . After calculating the
differential equations for u, γ, introduce a further rescaling of time by multiplying
the vectorfield by θ∗ cos2 u. Retaining the prime to denote differentiation with
respect to the new time variable one finds
r0 = θ∗ vr cos2 u
(12)
v 0 = θ∗ (G(u) − 21 v 2 cos2 u − 2r cos2 u) = θ∗ ( 14 γ 2 + 21 v 2 cos2 u − G(u))
u0 = γ
γ 0 = 4 Gu − 21 θ∗ vγ cos2 u + 4 sin u cos u(v 2 + 2r)
where
(13)
G(u) = cos2 u W (u)
1
m3
m3
W (u) =
+
+
ρ12
ρ13
ρ23
ρ212 = 1 + cos(θ∗ sin u)
ρ213 = A sin2 (θ∗ sin2 ( 21 (u + π2 )))
ρ223 = A sin2 (θ∗ sin2 ( 21 (u − π2 ))).
In deriving the differential equations, the energy relation
(14)
1 2
2v
cos2 u + 81 γ 2 − G(u) = rh cos2 u
has been used. It has been assumed that h = −1. Because of the scaling symmetry,
this is no loss of generality.
The Newtonian potential function W (u) has singularities at u = ± π2 corresponding to the double collision singularities (see figure 2). One can show that W (u) is
6
RICHARD MOECKEL
W
10
G
1
1

2
5
π

2
π
- 
2
u
-π
π
- 
2
W
20
G
4
10
3
π
- 
2
π

2
u
-π
π
- 
2
W
100
π

2
π
u
π

2
π
u
π

2
π
u
G
30
28
50
26
π
- 
2
π

2
u
-π
π
- 
2
Figure 2. Newtonian potential W (u) and regularized potential
1
G(u) for m3 = 10
, 1, 10
convex on (− π2 , π2 ) with second derivative Wuu (u) > 0. There is a unique critical
point at u = 0 corresponding to Euler’s collinear central configuration. On the
interval (0, π2 ), Wu (u) > 0. The regularized potential function G(u), on the other
hand, extends analytically to the double collisions to give an analytic function on
the entire real line of period π. G(u) has critical points at u = 0, π2 (and their
translates by integer multiples of π) with u = 0 giving its maximum.
The differential equations (12) represent the collinear three-body problem for
configurations with m3 in the middle, with triple collision blown-up and double
collisions regularized. The variable u describing the shape need not be confined to
the interval [− π2 , π2 ]. As u ranges over the real axis the shape variable θ oscillates
between the double collisions at ±θ∗ and the mass m3 bounces back and forth
between m1 and m2 .
The proof presented below makes use of the symmetry of the regularized potential
function G(u) to construct a symmetrical periodic solution as in figure 1. Figure 3
shows the periodic orbit with m3 = 1 in the (u, r) and (u, v) planes. The first
quarter of the orbit is a segment running from the vertical line u = 0 to the line
u = π2 with v = 0 at both endpoints. Then, using the symmetry of the vectorfield,
it follows that one can piece together the rest of the orbit by reflection through the
COLLINEAR PERIODIC ORBITS
r
7
v
2.8
0.2
π

2
2.7
π
3π

2
u
2π
-0.2
π

2
π
3π

2
u
2π
Figure 3. Symmetrical periodic solution viewed in the (u, r) and
(u, v)-planes. The orbit can be reconstructed from the segment
between u = 0 and u = π2 by reflection and translation.
line u = π2 followed by horizontal translation by π. The existence of the required
segment will be proved by a topological shooting argument.
3. A Wazewski Set
Consider the system (12) on the manifold of fixed energy h = −1. The goal of
this section is to construct a Wazewski set for the flow on this three-dimensional
manifold.
A Wazewski set for a flow φt (x) on a topological space X is a subset W ⊂ X
satisfying technical hypotheses which guarantee that the time required to exit W
depends continuously on initial conditions [11, 2]. To formulate these hypotheses,
let W 0 be the set of points in W which eventually leave W in forward time, and
let E the set of points which exit immediately:
W 0 = {x ∈ W : ∃t > 0, φt (x) ∈
/ W}
E = {x ∈ W : ∀t > 0, φ[0,t) (x) 6⊂ W}.
Clearly, E ⊂ W 0 . Given x ∈ W 0 define the exit time
τ (x) = sup{t ≥ 0 : φ[0,t) ⊂ W}.
Note that τ (x) = 0 if and only if x ∈ E. The appropriate hypotheses which
guarantee continuity of τ are [2]:
a. If x ∈ W and φ[0,t] ⊂ W, then φ[0,t] ⊂ W
b. E is a relatively closed subset of W 0
The choice of the set W is motivated by the shooting argument outlined at the
end of the last section. Let
W = {(r, v, u, γ) : (14) holds, r ≥ 0, v ≤ 0, 0 ≤ u ≤
π
2,γ
≥ 0}.
To visualize W one can use coordinates (r, v, u) on the energy manifold, using the
energy equation to find γ. The energy manifold projects to the solid region
(15)
r + 21 v 2 ≤ W (u).
Then W appears as in figure 4. The upper surface in the figure, where equality
holds in (15) corresponds to γ = 0. The figure shows several other important
features which will be explained in due course.
8
RICHARD MOECKEL
Figure 4. The Wazewski set W and a sketch of the existence proof .
Note that for any orbit segment in W the projection to the (u, r)-plane will lie
in the vertical strip over [0, π2 ] (see figure 3). Moreover, the curve will move in a
south-easterly direction, with u non-decreasing and r non-increasing.
The rest of this section is devoted to proving:
Theorem 1. W is a Wazewski set for the flow on the energy manifold.
The verification of defining property a is immediate since W is a closed subset
of R4 . To check property b, one must first identify the subsets W 0 , E. It turns out
that every solution beginning in W eventually leaves, with one exception — the
Eulerian triple collision solution mentioned in the introduction. Recall that this is
the simple homothetic solution obtained by putting m3 at the midpoint of m1 , m2
and releasing the masses with zero initial velocities.
To describe it more precisely, first note that there
p is a unique equilibrium point in
W at P = (r, v, u, γ) = (0, −v0 , 0, 0) where v0 = 2G(0). This corresponds to triple
collision with limiting shape u = 0, the Eulerian central configuration. As shown
by McGehee [8], the equilibrium point P is hyperbolic with two-dimensional stable
COLLINEAR PERIODIC ORBITS
9
manifold and one-dimensional unstable manifold. Indeed, using the coordinates
(v, u, γ) as local coordinates, one finds that the linearized differential equations at
the restpoint have matrix
 ∗

−θ v0
0
0
 0
0
1 .
0
4(Guu (0) + v02 ) 21 θ∗ v0
Using the fact that v0 > 0 and Guu (0) + v02 = Guu (0) + 2G(0) = Wuu (0) > 0
one finds that the eigenvalues are λ1 = −θ∗ v0 < 0 and two other real eigenvalues
λ2 < 0, λ3 > 0. The eigenvectors are (1, 0, 0), (0, 1, λ2 ), (0, 1, λ3 ) respectively. The
first stable eigenvector is tangent to the Eulerian homothetic orbit H = {(r, v, u, γ) :
u = γ = 0}. This orbit forms one of the edges of the Wazewski set W (the curved
edge with the arrow in figure 4). Note that the other stable eigenvector (0, 1, λ2 )
points out of W since any scalar multiple of it which has u > 0 must have γ < 0.
It follows that H ∩ W = W s (P ) ∩ W.
It is clear that any initial condition in H ∩ W remains there for all t ≥ 0 and
converges to P . However, these are the only orbits which remain in W:
Lemma 1. W 0 = W \ H.
Proof. Let x0 = (r0 , v0 , u0 , γ0 ) ∈ W. As long as the corresponding solution φt (x0 )
remains in W, one has bounds 0 ≤ r(t) ≤ r0 and 0 ≤ u(t) ≤ π2 . The energy
equation shows that γ(t)2 ≤ 8G(u(t)) so γ(t) is also bounded above and below.
There is an upper bound v(t) ≤ 0 but the energy equation does not provide a fixed
lower bound. There is a lower bound on the derivative, however:
v 0 ≥ −θ∗ G(u).
Then it follows from standard existence theorems that φt (x0 ) continues to exist as
long as it remains in W.
Now suppose x0 ∈ W \ H. It will be shown that φt (x0 ) eventually leaves W. If
u0 = 0 then u0 (0) = γ0 > 0 since x0 ∈
/ H. It follows that for every t0 > 0, u(t0 ) > 0.
Thus it suffices to consider initial conditions with u0 > 0.
Fix any u0 > 0 and let Wu0 = {x ∈ W : u ≥ u0 }. Note that since u(t) is
non-decreasing along orbits in W, Wu0 is positively invariant relative to W. It
will be shown below that there exist constants c0 > 0 and d0 > 0 such that for
every x ∈ Wu0 either γ ≥ c0 or γ 0 ≥ d0 . This will be enough to show that
φt (x0 ) must eventually leave W. To see this, write Wu0 = Wu+0 ∪ Wu−0 where
Wu+0 = {x ∈ Wu0 : γ ≥ c0 } and Wu−0 = {x ∈ Wu0 : 0 ≤ γ ≤ c0 }. Then since
γ 0 ≥ d0 > 0 in Wu−0 , it follows that an orbit segment can remain there for time
c0 /d0 , at most. Furthermore Wu+0 is positively invariant relative to Wu0 . Finally,
an orbit can remain in Wu+0 for at most time π/(2c0 ) since u0 = γ ≥ c0 . Thus every
orbit beginning in Wu0 eventually leaves W, as required.
It remains to construct c0 > 0, d0 > 0 such that either γ ≥ c0 or γ 0 ≥ d0 for all
x ∈ Wu0 . First note that for u = π2 , the energy equation gives γ 2 = 8G( π2 ) > 0. If
p
the constant c0 is chosen less than 8G( π2 ) then γ ≥ c0 will hold for u = π2 . On
the other hand, if u0 ≤ u < π2 , the equation for γ 0 can be written
γ 0 = 4Gu (u) − 12 θ∗ vγ cos2 u + tan u(8G(u) − γ 2 ) ≥ 4Gu (u) + tan u(8G(u) − γ 2 )
10
RICHARD MOECKEL
since v ≤ 0 and γ ≥ 0 in W. Now
4 cos3 u
Wu (u) > 0
sin u
on [u0 , π2 ]. Let k > 0 be its minimum value on this interval. Then one has
4Gu (u) cot u + 8G(u) =
γ 0 ≥ tan u(k − γ 2 ) ≥ tan u0 (k − γ 2 ).
p
p
Taking c0 = min( k/2, 8G( π2 )) and d0 = k tan u0 /2 completes the proof.
QED
To find the immediate exit set E one must examine the boundary points of W
(see figure 4). It is convenient to distinguish two subsets of the boundary. Let
x = (r, v, u, γ) and let
B1 = {x ∈ W : u =
π
2}
B2 = {x ∈ W : v = 0, 2r cos2 u ≤ G(u)}.
Note that B1 and B2 are closed subsets of W (shaded in figure 4). Therefore
the following lemma completes the verification of hypothesis b and the proof the
theorem 1.
Lemma 2. The immediate exit set of W is E = B1 ∪ B2 .
p
Proof. Let x ∈ B1 . Since u = π2 and u0 = γ = 8G( π2 ) > 0, x is clearly an
immediate exit point. Thus B1 ⊂ E.
Next consider a point in x ∈ B2 . Points with u = π2 are already known to be in
E, so one may assume 0 ≤ u < π2 . By definition, v = 0 and so one has
v 0 = θ∗ (G(u) − 2r cos2 u) ≥ 0.
If 2r cos2 u < G(u) then v 0 > 0 and x is an immediate exit point. On the other hand
if 2r cos2 u = G(u), one has v = v 0 = 0 and one finds that the second derivative
reduces to
v 00 = θ∗ γ(Gu (u) + 4r cos u sin u).
Furthermore the energy equation simplifies to γ 2 = 4G(u) > 0. Now the expression
in parentheses is positive unless u = 0. Therefore if 2r cos2 u = G(u) and 0 < u < π2 ,
one has v = v 0 = 0 and v 00 > 0 and again x is an immediate exit point. Finally if
u = v = 0 and 2r cos2 u = G(u) one has v = v 0 = v 00 = 0. The third derivative is
found to be
v 000 = θ∗ (Guu (0) + 2G(0)) = θ∗ Wuu (0) > 0.
Again, x is an immediate exit point and B2 ⊂ E.
To complete the proof, it remains to show that there are no other immediate
exit points. Suppose, for the sake of contradiction, that x0 ∈ W is an immediate
exit point which is not in B1 ∪ B2 . Rather than considering all of the various faces,
edges, and corners of the boundary it is easier to consider the logically possible
ways that x0 could exit and to rule them out.
First, one could exit by having r0 = 0 but r(t) < 0 for small positive times.
However, this is impossible because {r = 0} is the invariant triple collision manifold.
Next, one might have u0 = 0 and u(t) < 0 for small positive times. Clearly this
requires u0 (0) = γ0 ≤ 0 and since x0 ∈ W this means γ0 = 0. But u0 = γ0 = 0
defines Euler’s orbit H and points of H are certainly not immediate exit points.
A third possibility is that v0 = 0 and then v(t) increases to become positive.
This forces v 0 = θ∗ (G(u) − 2r cos2 u) ≥ 0 and so x0 ∈ B2 .
COLLINEAR PERIODIC ORBITS
11
The last possibility is γ0 = 0 with γ(t) becoming negative. Since u0 = γ0 = 0
defines H and u0 = π2 represents B1 , one may assume 0 < u0 < π2 . In this case, it
follows from the proof of lemma 1 that there are positive constants c0 , d0 such that
γ 0 ≥ d0 > 0 whenever γ < c0 . In particular, this applies to points with γ0 = 0 and
so this mode of exiting is also impossible. This completes the proof.
QED
4. The Shooting Argument
To construct symmetric periodic orbits, it suffices to show that there is an initial
condition with u = v = 0 which can be followed across W to an exit state with
u = π2 , v = 0. Let
S = {(r, v, u, γ) ∈ W : u = v = 0}
T = {(r, v, u, γ) ∈ W : u = φ/2, v = 0}.
S and T are two of the edges in the boundary of the three-dimensional Wazewski
set W
p(shown as bold vertical lines in figure 4). Along S one has 0 ≤ r ≤ G(0) and
γ = 8(W (0) − r). Viewing r as a parameter along S one sees that the endpoint
with r = 0 lies in the triple collision manifold while the endpoint with r = W (0) is
a point of the Eulerian homothetic orbit H. Let
S0 = {(r, v, u, γ) ∈ W : u = v = 0, 0 ≤ r < G(0)}.
Then S0 ⊂ W0 , that is, all of these points eventually exit W through E. Since
W is a Wazewski set, the time required to reach E depends continuously on initial
conditions and so there is a continuous flow-defined map F : S0 → E. Now the
target set T is contained in E. It remains to show that F (S0 ) ∩ T 6= ∅.
Again taking r as a parameter along S0 , one sees that the part of S0 with
0 ≤ r ≤ G(0)/2 is contained in B2 ⊂ E. These points exit W immediately and so
the map F is the identity there. On the other hand, points of S0 with r > G(0)/2
will enter the interior of W and emerge elsewhere. The proof will be completed by
studying the behavior of points near the other end of S0 , that is, with r ≈ G(0).
By continuity of the flow these points will follow the Eulerian homothetic orbit H
down to a neighborhood of the Eulerian equilibrium point at P = (0, −v0 , 0, 0). It
follows from the lambda lemma that they will then follow a branch of W u (P ).
Now W u (P ) is a one-dimensional manifold contained entirely in the invariant
triple collision manifold r = 0. One of the two branches is contained in W. In fact
it is contained in W0 and so can be followed under the flow to the exit set E. The
next lemma describes where it exits:
Lemma 3. The branch of W u (P ) in W exits W at a point of the form (0, v, π2 , γ)
with v < 0.
In other words, having started with u ≈ 0 and v ≈ −v0 < 0 it reaches the double
collision at u = π2 before it reaches v = 0.
Using this lemma, one can conclude the shooting argument as follows. In the
last section it was shown that the immediate exit set consists of two pieces B1 and
B2 of the boundary of W. As shown in figure 4, B1 and B2 are two-dimensional
surfaces meeting along the edge T . The continuous map F takes points of S0 near
r = 0 to B2 \ T and points near r = G(0) to B1 \ T . It follows that there must exist
at least one intersection point Q ∈ F (S0 ) ∩ T . This completes the existence proof
for the symmetric periodic orbits.
12
RICHARD MOECKEL
Theorem 2. Let three positive masses m1 = m2 and m3 and a negative energy h
be given. Then there exists a symmetric periodic solution of the collinear three-body
problem with energy h and regularized double collisions of the following type. During
the first quarter period, the masses move from the Eulerian central configuration
with m3 at the midpoint of m1 , m2 to a double collision between m2 and m3 . At
the moment of double collsion the velocity of m1 is zero. The second quarter of the
orbit is the time-reverse of the first and the second half is the reflection of the first
half with the roles of m1 and m2 reversed.
The proof also shows that the moment of intertia I is decreasing on the first quarter
of the orbit, after which it increases, decreases and increases again during the other
quarter periods.
It only remains to prove lemma 3 about the branch of the unstable manifold
W u (P ).
Proof of Lemma 3. Consider the differential equations for u, v. Using the energy
relation and the fact that the unstable branch lies in the triple collision manifold,
r = 0, these can be written:
v 0 = θ∗ (G − 21 v 2 cos2 u)
u0 = γ
where γ 2 = 8(G(u) − 12 v 2 cos2 u). So, parametrizing the branch by the variable u,
one has
dv
θ∗ p
θ∗ p
2G(u) − v 2 cos2 u ≤
2G(u).
=
du
4
4
π
Now G(u) is decreasing for 0 ≤ u ≤ 2 (see figure 2). Hence G(u) ≤ G(0) and
θ∗ p
dv
θ∗ v0
≤
2G(0) =
du
4
4
p
where v0 = 2G(0). It follows that the change in v for 0 ≤ u ≤ π2 satisfies:
π 2 v0
θ∗ πv0
≤
< v0
8
16
since θ∗ ≤ π2 . Since the branch of W u (P ) begins near P where u = 0 and v = −v0 ,
it follows that it arrives at u = π2 before reaching v = 0 as claimed.
QED
∆v ≤
COLLINEAR PERIODIC ORBITS
13
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School of Mathematics, University of Minnesota, Minneapolis MN 55455
E-mail address: [email protected]